By using the fixed point theorem, the author study the existence of radial positive solutions for the boundary value problem of the Kirchhoff type biharmonic equation. When the nonlinear term f satisfies appropriate  conditions, the author proves that there is  at least one radial positive solution to the problem.

By using the stability and bifurcation theory of differential equation, we studied the dynamic properties of a class of reaction-diffusion models for plankton with a non-local competition. Firstly, we gave the conditions for the existence and stability of positive constant equilibrium solutions. Secondly, we gave the conditions for the occurrence of Hopf bifurcation and steady-state bifurcation at the positive constant equilibrium solution, and calculated the properties of these bifurcations. Finally, we explained the  theoretical analysis results through numerical simulation.

We considered a class of two-dimensional singularly perturbed reaction-diffusion problems with  discontinuous reaction terms. Firstly, using spatial contrastive  structural theory, boundary layer function methods, smooth seaming method, and asymptotic differential inequality methods, we constructed  asymptotic expansion of the problem with solutions for the internal layer and boundary layer up to the n order, where n was an arbitary constant. Secondly, we  proved the existence and local asymptotic stability of the solution with internal layer,  and constructed high-precision asymptotic expansion of the solution. Finally, we applied the obtained theoretical results  to a numerical example.

By using the bifurcation method, the author study the bifurcation behavior of nodal solutions for two-point boundary value problems of fourth-order Kirchhoff type beam equation. When M and f satisfy suitable conditions, the author obtain  the existence results of their nodal solutions.

By using the function decomposition of the central Morrey spaces with variable exponent and hierarchical estimation methods, and with the help of the boundedness of the parameterized Marcinkiewicz integral operators with rough kernels on the  Lebesgue spaces with variable exponent, we gave the boundedness on the central Morrey spaces with variable exponent.

We established a multi-stage dynamic model to reflect the  characteristics of product  word-of-mouth communication based on  the laws of product word-of-mouth communication. Firstly, using qualitative analysis and stability theory, we analyzed the existence and stability of the model’s equilibrium points, and gave the conditions for the existence of backward bifurcation. Secondly, the Lyapunov function was used to demonstrate the global asymptotic stability of  the model’s information-free equilibrium point and the positive equilibrium point when a=0.  Finally, we  found  that the model  underwent saddle-node bifurcation, Hopf bifurcation, B-T bifurcation, and degenerate Hopf bifurcation through numerical simulation. The results show  that the coupling effect between product buyers and active disseminators of product word-of-mouth has a significant effect on the dynamic process of word-of-mouth communication.

We discussed the existence of radial symmetric solutions of a Kirchhoff equation of  nonlinearity with gradient term on an annulus in R^N(N≥2) by using the Leray-Schauder fixed point theorem. Under the nonlinearity satisfied certain  conditions which allowed the nonlinearity might be superlinear growth of any order on unknown function term, and quadratic growth on the gradient term of unknown function, the existence results of radial symmetric solutions were obtained.

By using the definition of sub-linear expectation and corresponding  inequalities, we gave the strong limit theorems for weighted sums of END sequence under sub-linear expectations,  and further studied the Marcinkiewicz-Zygmund type strong law of large numbers and Strassen-type invariance principle.

Aiming at the modeling problem of Z-valued time series data with volatility, we proposed a Z-valued Taylor-Schwert generalized autoregressive conditional heteroscedasticity model based on Poisson distribution. Firstly, some statistical properties of the model were derived. Secondly, the unknown parameters in the model were estimated by using the condition maximum likelihood estimation method, and the asymptotic properties of estimators were proved. Thirdly, numerical simulations were conducted to demonstrate the  performance of the estimation method. Finally, a real daily stock return data was considered, and the superiority of the proposed model over existing models was proved through analysis of the fitting results of the data.

We proposed a  two-level algorithm based on the virtual element discretization for both  steady-state and time-dependent PNP (Poisson-Nernst-Planck) equations. This algorithm first decoupled and linearized the PNP equations using the linear virtual element solution, and then solved them in the quadratic virtual element space. Compared with the commonly used Gummel algorithm for solving the PNP equations, this algorithm could accelerate the solving speed. Numerical experimental results, including both 
 steady-state and time-dependent PNP equations, show that compared with the Gummel algorithm with the linear virtual element, the two-level algorithm has  higher accuracy, and consumes less CPU time and is more efficient at  comparable accuracy.

Aiming at  the challenges of high-dimensional medical image data in survival analysis, we proposed an accelerated failure trace regression model. The regression parameters were estimated by using Kaplan-Meier weighting and the Peaceman-Rachford algorithm. The numerical simulation results show that the estimation performance of the accelerated failure trace regression model is better than that of the traditional Lasso regression model. We apply the model to Alzheimer’s disease image data to further verify its effectiveness and practical value.

Aiming at the pricing problem of American options under the time-fractional Black-Scholes model, we  proposed an effective numerical solution method.  Firstly, the linear complementarity model satisfied by the American option was transformed into a nonlinear parabolic problem on a bounded domain by using the variable substitution and penalty method. Secondly,  the semi-implicit finite difference method was used to solve the problem, and the error results of the method and the non negativity proof of the solution were given. Finally, numerical experiments were used to verify the correctness and effectiveness of the proposed method.

We study the D(2)-vertex sum distinguishing total coloring problem of a tricyclic graph with non zero tree height by using analytic method, contradiction method, and the Combinatorial Nullstellensatz, and obtain an upper bound on the D(2)-vertex sum distinguishing total coloring for this type of graphs is Δ(G)+3.

By using the discharging method, we  study the local strict neighbor-distinguishing edge coloring problem of sparse graphs, and obtain that if G is a graph with maximum degree at most four and maximum average degree mad(G)<16/5,  then the local strict neighbor-distinguishing index of G is at most 10.

Aiming at the problem of safety hazards of insufficient bridge crack detection, we  proposed a bridge crack detection algorithm based on YOLOx-s, combined with a small unmanned aerial vehicle platform. Firstly, we added a residual hole convolution module in  the backbone to solve the problem of   large scale changes and complex backgrounds in drone images. Secondly, we added a coordinate attention mechanism module in  PANET to improve the detection rate of small targets. Finally, we replaced the loss function with Focal loss to enhance the learning of positive samples and improve the stability of the model.  The experimental results show that compared with the YOLOx-s algorithm, the proposed method improves detection accuracy by 3.72 percentage points. On embedded devices, this method has better accuracy than other mainstream algorithms and can achieve real-time detection, which can be better applied in bridge crack detection for unmanned aerial vehicle.

In order to reduce the risk of network intrusion, the author proposed an intelligent detection method for network intrusion nodes based on niche genetic algorithm. Firstly, aggregation processing was implemented for the attack behavior of network intrusion,  a two-person attack and defense game model was used to analyze the attack and defense status of the network. By comparing the utility strength of attack and defense, a comprehensive analysis of the network’s security was carried out. Based on the analysis results, the localization of the attack source was achieved through convolutional neural networks. Secondly, based on  rough set theory, the fitness function for network intrusion node detection was determined by using niche genetic algorithm. According to  the intelligent detection rules of network intrusion nodes, an intelligent detection model for network intrusion nodes was established to obtain the final detection results. Experimental results show that this method can effectively improve the accuracy of locating intrusion attack sources and detecting intrusion nodes. The macro F₁  score of the detection results of this method is greater than 0.96, indicating that this method can effectively achieve the design expectations.

Aiming at the problem of emergency resource allocation for natural disasters, we proposed a Q-learning algorithm based on dynamic Boltzmann Softmax (DBS) and dynamic exploration rate (DER) (DBSDER-QL).  Firstly, the DBS strategy was used to dynamically adjust the weights of action values, promoting stable convergence of the algorithm and solving the problem of excessive of  the maximum operator. Secondly, the DER strategy was used to improve convergency and stability of the algorithm, solving the problem of the fixed exploration rate Q-learning algorithm not fully converging to the optimal strategy in the later stage of  training. Finally,  the effectiveness of the DBS and DER strategies was verified by ablation experiments. Compared with 
 dynamic programming, the greedy algorithm, and traditional Q-learning algorithm, the experimental results show  that DBSDER-QL algorithm is significantly better than traditional methods in terms of total cost and computational efficiency, showing higher applicability and effectiveness.

Aiming at  the problem of low accuracy of traditional support vector machines (SVM), we proposed an LDBO-SVM model. Firstly, 
in order to solve the problem of uneven distribution of the initial solution of the original dung beetle optimization algorithm, the Logistic chaotic map was introduced into the algorithm to construct the LDBO algorithm. Secondly, the LDBO algorithm was used to optimize the internal penalty factor and kernel parameters of the traditional support vector machine, and the LDBO-SVM model was constructed. Finally, in order to verify the performance of LDBO-SVM model, LDBO-SVM model was compared with the 
improved SVM by using five other population intelligent optimization algorithms. The experimental results show that the accuracy of LDBO-SVM model reaches 94.53%, and  can accurately predict student achievement, providing assistance for  teachers to improve their teaching plans.

Aiming at the problem of traditional methods  relying too much on local features and neglecting global features in texture classification, we proposed a feature extraction method based on the combination of local and non-local patterns.  The method integrated two algorithms: local phase quantization and non-local binary patterns. Firstly, two algorithms were used to  extract feature  from the preprocessed image separately. Secondly, the feature histograms of the two methods were weighted and fused. Finally, texture classification was performed by using the chi-square distance and the nearest neighbor classifier. In order to validate the effectiveness of the proposed method, a dataset of Manchu Eight Banners flag images was constructed, and the algorithm was applied to the classification task of the dataset. Experimental results show that, compared to single algorithm, the new algorithm has higher classification accuracy and robustness on multiple datasets.

Aiming at the problem that the filling of missing values in time series usually relied on the predictions of existing data, and  the complexity and uncertainty of time series often led to errors in the prediction results. In order to ensure the effectiveness of data filling, we proposed a time series missing data filling method based on generalized center clustering. Firstly, we calculated the distance between objects and classes, as well as between classes, quantified the relative positional relationship between data points and cluster centers, and obtained the spatial relationship between data. Secondly, we used information bottleneck algorithms to cluster the generalization centers in space， dividing time series datasets containing missing data into the same class. Finally, we calculated the cluster radius, divided the outlier data generated by the generalized center clustering into usable and weakly usable randomly damaged data, set a fluctuation threshold, and compared the randomly damaged data within the fluctuation threshold with a string of the unified attribute values in the cluster, achieving the  missing data filling in the time series. The experimental results show that this method has high standardized mutual information and hit rate in the clustering process,  and  can ensure a data replenishment rate of over 80% when filling in missing data, indicating that this method can effectively improve the integrity of time series data.

Aiming at the problems of insufficient acoustic  feature extraction under sample scarcity conditions, we proposed a small sample sound event detection method based on channel  and spatial compression.  The method constructed a dual compression attention mechanism to screen features  in channel dimension and achieved  feature focusing in the spatial dimension, effectively improving the feature discrimination ability of the prototype network in small sample scenarios. The experimental results show that F₁-score  of the method on the dataset DCASE (detection and classification of acoustic scenes and events) reaches 66.84%, an improvement of 4.11 percentage points compared to the prototypical network, providing more reliable technical  support for practical applications such as wildlife monitoring and ecological environment assessment.

In order to  improve efficiency of road traffic, by using  the minimum traffic navigation  cloud sacle  given number of road nodes and vehicles, we  proposed a performance analysis model based on target map and cloud center related parameters. Firstly, we analyzed  the target map and the relevant parameters of the cloud platform that affected  navigation performance.  Secondly, we  considered the number of road nodes and vehicles, as well as virtual machines of  different scales and frequencies, and conducted three sets of experiments.  Finally, we  verified the effectiveness  of the model by using real datasets. The experimental results show that the proposed performance analysis model can accurately calculate the minimum scale of  navigation cloud that satisfies the constraint conditions.

Aiming at the problem that uneven distribution of nodes led to  incomplete network coverage, the uneven position and density of nodes increased the complexity of deployment, and the search for the optimal node was prone to getting stuck in local optimal solutions in a multi hop wireless network, the author proposed  a non-uniform node deployment algorithm for  multi hop wireless network based on ant colony algorithm. Firstly, the author obtained the minimum weighted distance decision variable to reduce the transmission distance between Sink nodes and various sensors. Secondly, the author calculated the energy consumption of nodes, minimized node network loss, constructed a node deployment optimization model, and introduced compromise planning method to expand the multi-objective model into a single objective processing. Finally, the author introduced ant colony algorithm to solve the model, which could effectively traverse the potential solution space and quickly find the optimal deployment plan. The experimental results show that the  network coverage of proposed algorithm is 97%, with a maximum energy consumption of only 1.56×10^-7 J, which can effectively reduce network energy consumption. The survival cycle can reach up to 1 500 rounds, and the optimal node deployment plan can be obtained.

We used disturbance observer to study the adaptive sliding mode synchronization of permanent magnet synchronous motor (PMSM) fractional-order chaotic system. We considered uncertainties and unknown external disturbances through the fractional-order disturbance observer and verified the obtained conclusion through numerical simulation. The results show that the master-slave system of the PMSM chaotic system achieves sliding mode synchronization under  appropriate disturbance observer, sliding mode function, controller and the adaptive law.

In order to reduce the maximum temperature of the electronic components on printed circuit board (PCB) and optimize the scheme of thermal design. We constructed a mathematical model of the steady-state temperature field of electronic components on PCB by using the micro-element body heat balance method according to the principle of heat transfer, and verified the effectiveness of the model  by using the thermal simulation software ICEPAK for electronic products. We made  discretization improvement based on the surface-simplex swarm evolution algorithm, and used the improved discrete surface-simplex swarm evolution (DSSSE) algorithm to adjust the layout of distributed electronic components on PCB. Simulation results show that the proposed algorithm can reduce the maximum temperature of the electronic components on the PCB and has a fast convergence speed.

Aiming at the problem of frequent earthquakes and other natural disasters, based on the research on distributed fiber-optic sensing technology and the sensing principle and basic theory of Michelson fiber-optic interferometer, we designed a vector seismic sensor. The sensitive part of the fiber-optic seismic sensor was mainly composed of elastic thin sheets, mass blocks, circular rings and sensing fiber-optic  components. We conducted strain effect analysis of the sensor and designed overall scheme of the system. The signal generator was used to simulate earthquakes with the signal amplitude of  1 and the frequencies of 300,500 Hz, respectively. The results show that the composite materials are ideal materials for the vibrating thin sheet, which can significantly improve the sensitivity of the system and reduce the system errors. The detection sensitivities are 0.08 g and 0.3 g, respectively. The amplitude of the seismic signal is basically consistent with the signal amplitude detected by the sensor, and it has a good frequency response, which can be used for detecting seismic waves.

We characterized Na_0.52Bi_0.48-2xLi_xSr_xTi_0.99Mg_0.01O_3-δ（x=0,0.005,0.010,0.015) polycrystalline oxygen ion conductors prepared at a sintering temperature of 1 000 ℃ by using X-ray diffraction (XRD), scanning electron microscopy (SEM), Raman spectrum, and AC impedance spectroscopy to study the effect of the co-doping concentration of Li⁺ and Sr²⁺ on the crystal structure, microstructure and electrochemical properties of the series of the samples. The results show that with increase of co-doping concentration of Li⁺ and Sr²⁺, no obvious impurity peaks were observed in all samples, and the unit cell parameters and unit cell volumes first decreases and then increases with the increase of doping concentration. When x=0.010, the grain conductivity reaches its maximum value, which is about twice that of the undoped sample. When x=0.010, the apparent grain-boundary conductivity and the macroscopic grain-boundary conductivity of the sample reach their optimal values, which are superior to other doped samples.

We studied the chiral transition mechanism of the bis-α-alanine chelating Ca(Ⅱ) complex (α-Ala)₂→Ca(Ⅱ)  by using the M06-2X and MN15 methods of density functional theory (DFT). The results show that the chiral transition of (S-α-Ala)₂→Ca(Ⅱ) undergoes  two processes:  Firstly,  one of the α-Ala in (S-α-Ala)₂→Ca(Ⅱ) isomerizes from the S-type to the R-type to obtain R-allo-α-Ala→Ca(Ⅱ).  Secondly,  the S-α-Ala in R-allo-α-Ala→Ca(Ⅱ) isomerizes again to R-α-Ala to obtain (R-α-Ala)₂→Ca(Ⅱ).   The free energy barrier for the (S-α-Ala)₂→Ca(Ⅱ) isomerization to the R-allo-α-Ala→Ca(Ⅱ) tachycritical step under the implicit aqueous solvent  is 221.5 kJ/mol from the transition state where the H proton migrates from the chiral C atom to the  N atom， the free energy barrier drops to 93.1 kJ/mol in dominant aqueous solvent. The free energy barrier for the R-allo-α-Ala→Ca(Ⅱ) isomerization to (R-α-Ala)₂→Ca(Ⅱ) tachycritical step is 233.8 kJ/mol for the implicit aqueous solvent  from the transition state where the H proton  migrates from the chiral C atom to the  N atom, the free energy barrier  drops to 116.7 kJ/mol in the dominant aqueous solvent. Therefore,  the chiral transition rate of (S-α-Ala)₂→Ca(Ⅱ) is slow in physiological environments,  and   bis-α-Ala chelate calcium can be safely used for supplementation of living organisms with elemental calcium and α-Ala.

We established an expanded energy level system model of 2-(2′-hydroxyphenyl)benzothiazole (HBT) molecule and proposed corresponding numerical simulation algorithm for the dual-wavelength fluorescence phenomenon of organic molecule HBT in specific polar solutions， and verified the accuracy and effectiveness of the proposed algorithm.  The simulation results are similar to the experimental results, indicating that there is a competitive relationship between the fluorescence of HBT molecules at different wavelengths in specific  polar solutions. 

The NaBi(MoO₄)₂ electrode was prepared by the pressed sheet method,  and the material was modified by doping with Y³⁺. The morphology,  structure and electrochemical properties of the electrodes were characterized by X-ray diffraction (XRD), scanning electron microscope (SEM), transmission electron microscope (TEM), ultraviolet-visible-diffuse reflectance spectrum (UV-Vis-DRS) and electron spin resonance (ESR),  and the performance and reaction mechanism of the photocatalytic degradation of sildenafil were also investigated. The experimental results show that the NaBi(MoO₄)₂-based electrode  has a lamellar structure and high purity,  with the size ranging from about 37.5 nm to 24.6 nm. In the optimal electrolyte Na₂SO₄ solution with a concentration of 0.5 mol/L  and the optimal bias voltage of 1.5 V, sildenafil  can be completely removed after 70 min.